Theorem imai 4960

Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
imai	⊢ ( I “ 𝐴) = 𝐴

Proof of Theorem imai

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfima3 4949	. 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2		df-br 3983	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3		vex 2729	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	ideq 4756	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5	2, 4	bitr3i 185	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
6	5	anbi2i 453	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦))
7		ancom 264	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
8	6, 7	bitri 183	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
9	8	exbii 1593	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
10		eleq1 2229	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
11	3, 10	ceqsexv 2765	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
12	9, 11	bitri 183	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦 ∈ 𝐴)
13	12	abbii 2282	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴}
14		abid2 2287	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
15	1, 13, 14	3eqtri 2190	1 ⊢ ( I “ 𝐴) = 𝐴

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  ⟨cop 3579   class class class wbr 3982   I cid 4266   “ cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  rnresi  4961  cnvresid  5262  ecidsn  6548